INVESTIGATION OF BOUNDARY LAYER FOR A SECOND ORDER EQUATION UNDER LOCAL AND NON LOCAL BOUNDARY CONDITIONS

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1 ي J Basic Appl Sci Rs TtRoad Publication ISSN Journal of Basic and Applid Scintific Rsarch wwwttroadcom INVESTIGATION OF BOUNDARY LAYER FOR A SEOND ORDER EQUATION UNDER LOAL AND NON LOAL BOUNDARY ONDITIONS S Ashrafi* N Aliv Dpartmnt Of scinc Islamic Azad Univrsit Maraghh BranchIran Facult of Applid Mathmatics and brnticsbau Stat Univrsit ZKhalilov str3az48azrbaijan ABSTRAT Aris or absnc of a boundar lar a prturbd ordinar linar diffrntial quation of scond ordr is invstigatd in th papr In th first two ampls a boundar strip ariss in th lft or right and rspctivl In th third strip thr is no strip but in th fourth ampl it appars in th both nds Finall at th last ampl aris of a lar is invstigatd undr on non- local boundar condition KEYWORDS: prturbd quation; non-local boundar condition; adjoint quation; fundamntal solution; ncssar condition; boundar lar INTRODUTION It is nown that mathmatical modl of boundar lar aris is rducd to th problm for a diffrntial quation with cofficint of a highr ordr drivativ contains a small paramtr If in th boundar valu problm th conditions ar local and limiting stat of th solution of th problm whn a small paramtr tnds to zro dosn t satisf on of th givn conditions thn thr ariss a boundar lar at th point whr boundar conditions ar givn Prandtl 95 first studid this problm In squl mathmatical modls of othr natural phnomna wr rducd to ths problms Main invstigators of ths problms wr REO Mallr 99 and Tihonov 95 Duolan t al 98 said that this boundar valu problm rmains opn for th scond ordr ordinar diffrntial quation Hr at first an quation adjoint to th givn quation is constructd and thn fundamntal solution of this adjoint quation is found Th ncssar condition that satisfis arbitrar solution of th givn quation is dtrmind b mans of this fundamntal solution With th hlp of this ncssar condition th boundar conditions of th statd problm is localizd Th cas whn undr non-local condition th prturbd problm rmains opn is shown in th papr Duolan t al 98 B mans of ncssar conditions obtaind whil invstigating this problm th boundar conditions ar localizd and th plac of arising of boundar strip is dfind Up to now in invstigation of boundar valu problms is statd for th scond ordr quations Aliv 989 Jahanshahi Aliv t al 989 Pashav t al 995 and Aliv t al 9 it was succdd to obtain final rsult naml in this papr So for arising or not arising of a boundar lar th conditions ar rducd on data A problm for th nonlinar sstm of th scond ordr is considrd in Jahanshahi t al 4 Invstigation of Aris of a Boundar Lar for Scond Ordr Equation Undr Local onditions Now lt's considr th following problm: *orrsponding Author: S Ashrafi Dpartmnt of Mathmatics Islamic Azad Univrsit- Maraghh Branch Maraghh Iran siamaashrafi@iau-maraghhacir 75

2 Ashrafi and Aliv Obviousl th gnral solution of Eq is of th form: 3 Hr ar arbitrar constants ar th roots of th charactristic quation Of th from 8 4 Using conditions w dtrmin arbitrar constants containd in th gnral solution W put th valus Obtaind from th obtaind sstm to Eq3 and find prssions for th solution of problm Eqs and : 5 For dtrmining limit stat of this solution at first w dtrmin limit roots in Eq 4 Sinc lim 8 8 lim 8 lim lim Th from Eq5 w gt 6 Sinc Eq 6 satisfis th scond condition from Eq but dosn't satisf th first on a boundar lar ariss at th nd Thorm: For boundar valu problm Eqs and a boundar lar ariss on th boundar 75

3 J Basic Appl Sci Rs Now considr th scond boundar valu problm: 7 8 Gnral solution of Eq7 is of th form: 9 B mans of boundar conditions 8 w dtrmin th arbitrar constants and containd in Eq9: Thn th solution of problm Eqs6 and 8 is obtaind on th form: Whnc for w find Sinc this function satisfis th first condition from 8 but dosn't satisf th scond on a boundar lar ariss at th right nd Thorm For boundar valu problm Eqs7 and 8 a boundar strip ariss on th boundar onsidr th following third problm: 3 W can asil show that gnral solution of Eq is of th from: 4 75

4 Ashrafi and Aliv Hr th first two trms is a gnral solution of a homognous quation corrsponding to Eq and th sum of two last trms is a particular solution of Eq Using conditions 3 w find constants and containd in th solution 4: Having substitutd th valus of th constants and in Eq4 for th solution of Eqs and 3 w gt th prssion: 5 Passing to limit as from Eq5 w find 6 Sinc th obtaind function 6 satisfis conditions 3 a boundar lar is not to b found anwhr Thorm 3 A boundar lar dosn't aris for Eqs and 3 At last w considr th following problm: 7 8 Gnral solution of Eq7 is of th form: 9 753

5 J Basic Appl Sci Rs From th sstms W dtrmin th constants and as And having substitutd thn in 9 for th solution of problm Eqs3 and 8 w gt th prssion: Whr Sinc this function dosn't satisf conditions 8 a boundar lar ariss at both nds Thorm 4 For th boundar valu Eqs7 and 8 a boundar lar ariss at both boundaris 3 Invstigation of Aris of a Boundar Lar for a Scond Ordr Equation Undr Non-Local Boundar ondition Dfinition of aris or absnc of a boundar lar undr non-local boundar conditions is not difficult Indd if limit stat of th solution of th givn problm dosn't satisf thn of on condition than thr ariss a boundar lar But it in difficult to dfin a boundar a which a boundar lar ariss In this connction w considr th following problm: 3 4 Solv this problm b ordinar mthod Gnral solution of Eq is of th form: Hr and ar arbitrar constants ar th roots of th charactristic quation 754

6 Ashrafi and Aliv 3 That is of th form: 3 For using th boundar condition w diffrntiat slt put it into and gt: If in w writ th found constants w gt an prssion for th solution of Eqs and in th form: 4 So that In ordr to pass to limit as in prssion 4 obtaind for th solution of th problm at first w dtrmin prssions for th roots of charactristic Eq3 as lim 3 lim onsquntl from 4 w gt: lim 6 Sinc th obtaind function 6 satisfis th first condition but dosn't satisf th scond condition from thn thr ariss a boundar lar for Eqs and But it is not mown on which boundar it ariss It is as to s that Vladimirov 976 th quation adjoint to Eq is of th form: z z z 3z g * Whr g is an arbitrar continuous function Th fundamntal solution of this adjoint quation is of th form Aliv and Jahanshahi 997: z 7 Hr t is Havisid's unit function ar th roots of th charactristic quation 755

7 J Basic Appl Sci Rs That ar of th form: 8 W can asil s that 3 9 Having scalar multiplid th obtaind fundamntal solution and its drivativ b th both sids of Eq and intgrating b part similar to Aliv and Jahanshahi and Aliv and Hossini 3 and Aliv and Ashrafi 9 w gt following ncssar conditions: 3 3 Passing to limit in ncssar Eq3 and Eq3 and givn boundar Eq w gt a sstm of four linar algbraic quations W solv this sstm and find : 3 4 Sinc th obtaind Eq7 dosn't satisf th last condition from 3 thn a boundar lar ariss at th point Thorm 3 In Eqs and with non-local boundar condition a boundar lar ariss at th point REFERENES Duolan EPMillr JJH& Schildrs LHA 98 Uniform Numrical Mthods for Problms with initial Boundar lars Bool Prss Dublim Vladimirov VS 976Equation of Mathmatical Phsics 3rd Naua Moscow in Russian [English transl of nd Russian d Marcl Dr Nw Yor 97] Aliv N & Ashrafi S 9 Invstigation of origin of boundar lar on a problm for a scond ordr ordinar linar prturbd diffrntial quation undr gnral non-local boundar condition Adv In Math Sci Jour AMSJ Aliv N & Jahanshahi M 997 Sufficint onditions for rduction of BVP Including a mid PDE with Non- local boundar conditions to Frdholm Intgral Equations INT JMath Educ Sci &Tch 8 No Aliv N & Jahanshahi M Sufficint of Poisson's Equation with global local and non-local boundar conditions INT J Math Educ Sci Tchnol 33 No

8 Ashrafi and Aliv Aliv N & S M Hossini 3 An Analsis of a Parabolic Problm with a Gnral Non-local and Global Supplmntar linar conditions- II Italian Journal of pur and Applid Mathmatics Jahanshahi M Aliv M & Hossini SM 9 An Analtic Mthod for Invstigation and Solving Two-Dimnsional Stad Stat Navir-Stos Equations I Southast Asian Bulltin of Mathmatics 33: Prandtl 95 Ubr Flussiglits - bowngung bi linr Ribng vrhandlungn III int Math Kongrsss Tubnr Lipzig Tihonov AN 95 Sstm of diffrntial quations containing small paramtrs at th drivativs Math sb 3 No 3 95 REO Mall Jr 99 Singular prturbation mthods for ordinar diffrntial quations Springr- Vrlag Aliv N and Sulimanov N 989 Boundar valu problms with small paramtr in boundar condition Bau Stat Univrsit Sb Nauch Trud Azrbaijan Bau Jahanshahi M Invstigation of boundar lars in a singular prturbation problm 3-th Annual Iranian Mathmatical onfrnc Thran Iran Sptmbr Aliv N & YMSalmanov YM 989 Rsarch of uniformal corrctnss of boundar valu problm for 4th ordr ODE Dp AZNIINTI 7 No 48 Pashav R T and Aliv N 995 Uniformal corrctnss of BVP for th scond ordr ODE Nws of Bau Stat Univrsit NI Mhri B and Jahanshahi M998 Linar Diffrntial Equation with Prturb Boundar condition Procdings of 9th Annual Iranian Mathmatical confrnc 8-3 March Thran Iran Jahanshahi M Mhri B Aliv N & Saai K 4 Asmptotic pansions of Priodic solutions for a singular prturbation problm including nonlinar dnamical sstm with two boundar lars Southast Asian Bulltin of Mathmatics